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Local existence and blow‐up criterion for the generalized Boussinesq equations in Besov spaces
Author(s) -
Qiu Hua,
Du Yi,
Yao Zheng'an
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2573
Subject(s) - mathematics , extension (predicate logic) , laplace operator , fractional laplacian , type (biology) , mathematical analysis , simultaneous equations , magnetohydrodynamics , boussinesq approximation (buoyancy) , differential equation , magnetic field , physics , geology , paleontology , convection , natural convection , quantum mechanics , computer science , rayleigh number , programming language , thermodynamics
In this paper, we consider the three‐dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian − Δ in the usual Boussinesq equations by a fractional Laplacian ( − Δ) α . We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin‐type criteria for Navier–Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley & Sons, Ltd.